for some definitions of 'almost', especially the vague human ones, there are very surprising results (well, not for people with CS education, but for the other ones). e.g. 2-SAT vs 3-SAT, hamiltonian path (visit all nodes on a graph) vs eulerian path (visit all edges), shortest path vs longest path, etc. like i said, the 'almost' is debatable, but for the untrained eye the problems are very similar.
Face/shape recognition comes to mind, where inexact graph matching would be useful, but are there any others?
I'm not trolling, just interested in graph matching in general as amateur. I have this feeling that there should be many more applications of these techniques, but amusingly, in practice there's always a more specialized, non matching algorithm used. (probably because of matching's algorithm complexity which doesn't scale for larger problems...)
Take your problem in Constraint Programming / Mixed Integer Programming / SAT / SMT, generate a parse tree, turn that parse tree into a graph, and find symmetries of that graph.
If done in the right way, symmetries of the graph are symmetries of the original problem, and knowledge of these symmetries can be used to avoid redundant symmetric search. This is done all the time in combinatorial search systems.
One problem for many other real-world problems is it is much more common to want "almost symmetries" (for various definitions of almost symmetry). It turns out this is a much harder problem, and algorithms for "pure" graph isomorphism can't be easily modified, to solve the 'almost symmetry' problem.